Some remarks on finite element approximation of multiple eigenvalues

In this paper we investigate the behavior of the finite element approximation of multiple eigenvalues in presence of eigenfunctions with different smoothness. We start from a one-dimensional example presented in the Handbook of Numerical Analysis by Babuška and Osborn and extend it to higher order approximation and to two dimensions, confirming that the different regularities of the eigenfunctions are well seen in the numerical computations. Then we discuss a mixed formulation corresponding to the one-dimensional example. It turns out that the regularity properties of the eigenfunctions are not well separated in this particular example, since the estimates have to take into account both components of the solution.